HARMONIC MORPHISMS AND HERMITIAN STRUCTURES ON EINSTEIN 4-MANIFOLDS

1992 ◽  
Vol 03 (03) ◽  
pp. 415-439 ◽  
Author(s):  
JOHN C. WOOD
Keyword(s):  
Riemann Surface ◽  
Critical Points ◽  
Minimal Surfaces ◽  
Hermitian Structure ◽  
Harmonic Morphisms ◽  
Harmonic Morphism ◽  
Degeneracy Condition ◽  
Hermitian Structures

We show that a submersive harmonic morphism from an orientable Einstein 4-manifold M4 to a Riemann surface, or a conformal foliation of M4 by minimal surfaces, determines an (integrable) Hermitian structure with respect to which it is holomorphic. Conversely, any nowhere-Kähler Hermitian structure of an orientable anti-self-dual Einstein 4-manifold arises locally in this way. In the case M4=ℝ4 we show that a Hermitian structure, viewed as a map into S2, is a harmonic morphism; in this case and for S4, [Formula: see text] we determine all (submersive) harmonic morphisms to surfaces locally, and, assuming a non-degeneracy condition on the critical points, globally.

HARMONIC MORPHISMS FROM EINSTEIN 4-MANIFOLDS TO RIEMANN SURFACES

2003 ◽  
Vol 14 (03) ◽  
pp. 327-337 ◽  
Author(s):  
MARINA VILLE
Keyword(s):  
Riemann Surface ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Complex Structure ◽  
Hermitian Structure ◽  
Harmonic Morphisms ◽  
Harmonic Morphism ◽  
The Real ◽  
Real Analyticity

If M and N are Riemannian manifolds, a harmonic morphism f : M → N is a map which pulls back local harmonic functions on N to local harmonic functions on M. If M is an Einstein 4-manifold and N is a Riemann surface, John Wood showed that such an f is holomorphic w.r.t. some integrable complex Hermitian structure defined on M away from the singular points of f. In this paper we extend this complex structure to the entire manifold M. It follows that there are no non-constant harmonic morphisms from [Formula: see text] or [Formula: see text] to a Riemann surface. The proof relies heavily on the real analyticity of the whole situation. We conclude by an example of a non-constant harmonic morphism from [Formula: see text] to [Formula: see text].

scholarly journals KNOT SINGULARITIES OF HARMONIC MORPHISMS

2001 ◽  
Vol 44 (1) ◽  
pp. 71-85 ◽  
Author(s):  
Paul Baird
Keyword(s):  
Riemann Surface ◽  
Complex Surface ◽  
Subject Classification ◽  
Harmonic Morphisms ◽  
Singular Set ◽  
Harmonic Morphism ◽  
Complex Curves ◽  
Analytic Curve ◽  
Primary 57M25 ◽  
Straight Lines

AbstractA harmonic morphism defined on $\mathbb{R}^3$ with values in a Riemann surface is characterized in terms of a complex analytic curve in the complex surface of straight lines. We show how, to a certain family of complex curves, the singular set of the corresponding harmonic morphism has an isolated component consisting of a continuously embedded knot.AMS 2000 Mathematics subject classification: Primary 57M25. Secondary 57M12; 58E20

Harmonic Morphisms as Sphere Bundles Over Compact Riemann Surfaces

1997 ◽  
Vol 08 (07) ◽  
pp. 935-942
Author(s):  
Sigmundur Gudmundsson
Keyword(s):  
Riemann Surface ◽  
Homotopy Type ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Harmonic Morphisms ◽  
Sphere Bundle ◽  
Harmonic Morphism ◽  
Totally Geodesic ◽  
Compact Riemann Surfaces

We prove that the projection map of an orientable sphere bundle, over a compact Riemann surface, of any homotopy type can be realized as a harmonic morphism with totally geodesic fibres.

scholarly journals Harmonic morphisms applied to classical potential theory

2011 ◽  
Vol 202 ◽  
pp. 107-126
Author(s):  
Bent Fuglede
Keyword(s):  
Potential Theory ◽  
Unit Sphere ◽  
Harmonic Morphisms ◽  
Harmonic Morphism ◽  
Open Set ◽  
Superharmonic Functions ◽  
Radial Projection ◽  
Classical Potential ◽  
Classical Potential Theory

AbstractIt is shown that ifϕdenotes a harmonic morphism of type Bl between suitable Brelot harmonic spacesXandY, then a functionf, defined on an open setV ⊂ Y, is superharmonic if and only iff ∘ ϕis superharmonic onϕ–1(V) ⊂ X. The “only if” part is due to Constantinescu and Cornea, withϕdenoting any harmonic morphism between two Brelot spaces. A similar result is obtained for finely superharmonic functions defined on finely open sets. These results apply, for example, to the case whereϕis the projection from ℝNto ℝn(N > n ≥1) or whereϕis the radial projection from ℝN\ {0} to the unit sphere in ℝN(N≥ 2).

scholarly journals Embedded minimal surfaces of finite topology

2019 ◽  
Vol 2019 (753) ◽  
pp. 159-191 ◽  
Author(s):  
William H. Meeks III ◽  
Joaquín Pérez
Keyword(s):  
Asymptotic Behavior ◽  
Riemann Surface ◽  
Finite Number ◽  
Minimal Surface ◽  
Minimal Surfaces ◽  
Compact Riemann Surface ◽  
Finite Topology ◽  
Compact Boundary ◽  
Interior Points

AbstractIn this paper we prove that a complete, embedded minimal surface M in {\mathbb{R}^{3}} with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface {\overline{M}} with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic data on its conformal completion {\overline{M}}. In particular, we demonstrate that M is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of M.

scholarly journals Harmonic morphisms with one-dimensional fibres on conformally-flat Riemannian manifolds

2008 ◽  
Vol 145 (1) ◽  
pp. 141-151 ◽  
Author(s):  
RADU PANTILIE
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Harmonic Morphisms ◽  
Two Dimensional ◽  
Conformally Flat ◽  
Harmonic Morphism ◽  
One Dimensional ◽  
Dimension 2 ◽  
Real Analytic ◽  
Flat Riemannian Manifold

AbstractWe classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four (Theorem 3.1), and (2) between conformally-flat Riemannian manifolds of dimensions at least three (Corollaries 3.4 and 3.6).Also, we prove (Proposition 2.5) an integrability result for any real-analytic submersion, from a constant curvature Riemannian manifold of dimensionn+2 to a Riemannian manifold of dimension 2, which can be factorised as ann-harmonic morphism with two-dimensional fibres, to a conformally-flat Riemannian manifold, followed by a horizontally conformal submersion, (n≥4).

Harmonic morphisms as unit normal bundles¶of minimal surfaces

1999 ◽  
Vol 100 (3) ◽  
pp. 323-333
Author(s):  
Sigmundur Gudmundsson ◽  
Xiaohuan Mo
Keyword(s):  
Minimal Surfaces ◽  
Harmonic Morphisms ◽  
Normal Bundles ◽  
Unit Normal

BUBBLING LOCATION FOR SEQUENCES OF APPROXIMATE f-HARMONIC MAPS FROM SURFACES

2010 ◽  
Vol 21 (04) ◽  
pp. 475-495 ◽  
Author(s):  
YUXIANG LI ◽  
YOUDE WANG
Keyword(s):  
Riemannian Manifold ◽  
Riemann Surface ◽  
Critical Points ◽  
Homotopy Class ◽  
Blow Up ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Identity ◽  
Minimal Point ◽  
Closed Riemann Surface

Let f be a positive smooth function on a closed Riemann surface (M, g). The f-energy of a map u from M to a Riemannian manifold (N, h) is defined as [Formula: see text] and its L2-gradient is: [Formula: see text] We will study the blow-up properties of some approximate f-harmonic map sequences in this paper. For a sequence uk : M → N with ‖τf(uk)‖L2 < C1 and Ef(uk) < C2, we will show that, if the sequence is not compact, then it must blow-up at some critical points of f or some concentrate points of |τf(uk)|2dVg. For a minimizing α-f-harmonic map sequence in some homotopy class of maps from M into N we show that, if the sequence is not compact, the blow-up points must be the minimal point of f and the energy identity holds true.

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